On the Hopf Bifurcation in Control Systems with a Bounded Nonlinearity Asymptotically Homogeneous at Infinity

Abstract

The paper studies existence, uniqueness, and stability of large-amplitude periodic cycles arising in Hopf bifurcation at infinity of autonomous control systems with bounded nonlinear feedback. We consider systems with functional nonlinearities of Landesman Lazer type and a class of systems with hysteresis nonlinearities. The method is based on the technique of parameter functionalization and methods of monotone concave and convex operators. Academic Press

Pattern formation in parabolic equations containing hysteresis with diffusive thresholds

“…Here the first integral is the total mass of bacteria and P is the so-called Preisach operator [28] with the time dependent density function u; differential equations with the Preisach operator have been studied, for example, in [5,43,2,9,4,12,11,25,24,23]. Further, we replace the unknown functions f 1 and f −1 in system (2.1) by v = f 1 + f −1 (total mass of the two nutrients) and w = f 1 /(f 1 + f −1 ) − 1/2 (deviation of the relative concentration of the first nutrient from the value 1/2).…”

Generalized Hopf bifurcation in a class of planar switched systems

“…Another type of DIB related to Hopf bifurcation or generalized Hopf bifurcation concerns the appearance or disappearance of a periodic orbit [6,8,[10][11][12][13][14]. The papers [6,8] considered a case when an equilibrium point hits the line of discontinuity, then becomes unstable and creates periodic oscillations.…”

“…The papers [6,8] considered a case when an equilibrium point hits the line of discontinuity, then becomes unstable and creates periodic oscillations. In [10,11], a case of Hopf bifurcation in which the periodic orbits are generated from infinity was considered. Kupper et al [12,13] studied the generalized Hopf bifurcation for the case that the equilibria always stay in the smooth line of discontinuity and the bifurcating periodic orbit transversely crosses the line of discontinuity at least twice.…”

Generalized Hopf bifurcation emerged from a corner in general planar piecewise smooth systems

“…The recent work on GHB in planar PWSs with a line of discontinuity can be found in [15][16][17][18][19][20][21], etc. Almost all these studies are concerned with the case when the Jacobian matrix of each smooth subsystem has a pair of complex conjugate eigenvalues, and most of these Jacobian matrices are given in their Jordan normal forms.…”